.\"                                      Hey, EMACS: -*- nroff -*-
.TH LMDIF_ 3 "March 8, 2002" Minpack
.\" Please adjust this date whenever revising the manpage.
.SH NAME
lmdif_, lmdif1_ \- minimize the sum of squares of m nonlinear functions 
.SH SYNOPSIS
.B include <minpack.h>
.nh
.ad l
.HP 15
.BI "void lmdif1_ ( "
.BI "void (*" fcn ")(int *" m ", int *" n ", double *" x ,
.BI "double *" fvec ", int *" iflag ),
.RS 2
.BI "int *"      m        ,
.BI "int * "     n        ,
.BI "double *"   x        ,
.BI "double *"   fvec     ,
.br
.BI "double *"   tol      , 
.BI "int *"      info     ,
.BI "int *"      iwa      ,
.BI "double *"   wa       ,
.BI "int *"      lwa      );
.RE
.HP 14
.BI "void lmdif_"
.BI "( void (*" fcn ")(int *" m ", int *" n ", double *" x , 
.BI "double *" fvec ", int *" iflag ),
.RS 2
.BI "int *"      m        ,
.BI "int *"      n        ,
.BI "double *"   x        ,
.BI "double *"   fvec     ,
.br
.BI "double *"   ftol     ,
.BI "double *"   xtol     ,
.BI "double *"   gtol     ,
.BI "int *"      maxfev   ,
.BI "double *"   epsfcn   ,
.BI "double *"   diag     ,
.BI "int *"      mode     ,
.BI "double *"   factor   ,
.BI "int *"      nprint   ,
.BI "int *"      info     ,
.BI "int *"      nfev     ,
.BI "double *"   fjac     ,
.br
.BI "int *"      ldfjac   ,
.BI "int *"      ipvt     ,
.BI "double *"   qtf      ,
.br
.BI "double *"   wa1      ,
.BI "double *"   wa2      ,
.BI "double *"   wa3      ,
.BI "double *"   wa4      " );"
.RE
.hy
.ad b
.br
.SH DESCRIPTION

The purpose of \fBlmdif_\fP is to minimize the sum of the squares of
\fIm\fP nonlinear functions in \fIn\fP variables by a modification of
the Levenberg-Marquardt algorithm. The user must provide a subroutine
which calculates the functions. The Jacobian is then calculated by a
forward-difference approximation.

\fBlmdif1_\fP serves the same purpose but has a simplified calling
sequence.
.br
.SS Language notes
These functions are written in FORTRAN. If calling from
C, keep these points in mind:
.TP
Name mangling.
With \fBg77\fP version 2.95 or 3.0, all the function names end in an
underscore.  This may change with future versions of \fBg77\fP.
.TP
Compile with \fBg77\fP.
Even if your program is all C code, you should link with \fBg77\fP
so it will pull in the FORTRAN libraries automatically.  It's easiest
just to use \fBg77\fP to do all the compiling.  (It handles C just fine.)
.TP
Call by reference.
All function parameters must be pointers.
.TP
Column-major arrays.
Suppose a function returns an array with 5 rows and 3 columns in an
array \fIz\fP and in the call you have declared a leading dimension of
7.  The FORTRAN and equivalent C references are:
.sp
.nf
	z(1,1)		z[0]
	z(2,1)		z[1]
	z(5,1)		z[4]
	z(1,2)		z[7]
	z(1,3)		z[14]
	z(i,j)		z[(i-1) + (j-1)*7]
.fi
.br
\fIfcn\fP is the name of the user-supplied subroutine which
calculates the functions. In FORTRAN, \fIfcn\fP must be declared
in an external statement in the user calling
program, and should be written as follows:
.sp
.nf
  subroutine fcn(m,n,x,fvec,iflag)
  integer m,n,iflag
  double precision x(n),fvec(m)
  ----------
  calculate the functions at x and
  return this vector in fvec.
  ----------
  return
  end
.fi
.sp
In C, \fIfcn\fP should be written as follows:
.sp
.nf
  void fcn(int m, int n, double *x, double *fvec, int *iflag)
  {
  	  /* calculate the functions at x and return 
  	     the values in fvec[0] through fvec[m-1] */
  }
.fi
.sp
The value of \fIiflag\fP should not be changed by \fIfcn\fP unless the
user wants to terminate execution of \fBlmdif_\fP (or \fBlmdif1_\fP). In
this case set \fIiflag\fP to a negative integer.
.br
.SS Parameters for both \fBlmdif_\fP and \fBlmdif1_\fP

\fIm\fP is a positive integer input variable set to the number
of functions.

\fIn\fP is a positive integer input variable set to the number
of variables. \fIn\fP must not exceed \fIm\fP.

\fIx\fP is an array of length \fIn\fP. On input \fIx\fP must contain
an initial estimate of the solution vector. On output \fIx\fP
contains the final estimate of the solution vector.

\fIfvec\fP is an output array of length \fIm\fP which contains
the functions evaluated at the output \fIx\fP.
.br
.SS Parameters for \fBlmdif1_\fP 

\fItol\fP is a nonnegative input variable.  Termination occurs when
the algorithm estimates either that the relative error in the sum of
squares is at most \fItol\fP or that the relative error between
\fIx\fP and the solution is at most \fItol\fP.

\fIinfo\fP is an integer output variable. if the user has
terminated execution, \fIinfo\fP is set to the (negative)
value of iflag. see description of \fIfcn\fP. otherwise,
\fIinfo\fP is set as follows.

  \fIinfo\fP = 0  improper input parameters.

  \fIinfo\fP = 1  algorithm estimates that the relative error
in the sum of squares is at most \fItol\fP.

  \fIinfo\fP = 2  algorithm estimates that the relative error
between x and the solution is at most \fItol\fP.

  \fIinfo\fP = 3  conditions for \fIinfo\fP = 1 and \fIinfo\fP = 2 both hold.

  \fIinfo\fP = 4  \fIfvec\fP is orthogonal to the columns of the
Jacobian to machine precision.

  \fIinfo\fP = 5  number of calls to \fIfcn\fP has reached or
exceeded 200*(\fIn\fP+1).

  \fIinfo\fP = 6  \fItol\fP is too small. no further reduction in
the sum of squares is possible.

  \fIinfo\fP = 7  \fItol\fP is too small. no further improvement in
the approximate solution x is possible.

\fIiwa\fP is an integer work array of length \fIn\fP.

\fIwa\fP is a work array of length \fIlwa\fP.

\fIlwa\fP is an integer input variable not less than \fIm\fP*\fIn\fP +
5*\fIn\fP + \fIm\fP.
.br
.SS Parameters for \fBlmdif_\fP

\fIftol\fP is a nonnegative input variable. Termination
occurs when both the actual and predicted relative
reductions in the sum of squares are at most \fIftol\fP.
Therefore, \fIftol\fP measures the relative error desired
in the sum of squares.

\fIxtol\fP is a nonnegative input variable. Termination
occurs when the relative error between two consecutive
iterates is at most \fIxtol\fP. Therefore, \fIxtol\fP measures the
relative error desired in the approximate solution.

\fIgtol\fP is a nonnegative input variable. Termination
occurs when the cosine of the angle between \fIfvec\fP and
any column of the Jacobian is at most \fIgtol\fP in absolute
value. Therefore, \fIgtol\fP measures the orthogonality
desired between the function vector and the columns
of the Jacobian.

\fImaxfev\fP is a positive integer input variable. Termination
occurs when the number of calls to \fIfcn\fP is at least
\fImaxfev\fP by the end of an iteration.

\fIepsfcn\fP is an input variable used in determining a suitable
step length for the forward-difference approximation. This
approximation assumes that the relative errors in the
functions are of the order of \fIepsfcn\fP. If \fIepsfcn\fP is less
than the machine precision, it is assumed that the relative
errors in the functions are of the order of the machine
precision.

\fIdiag\fP is an array of length \fIn\fP. If \fImode\fP = 1 (see
below), \fIdiag\fP is internally set. If \fImode\fP = 2, \fIdiag\fP
must contain positive entries that serve as
multiplicative scale factors for the variables.

\fImode\fP is an integer input variable. If \fImode\fP = 1, the
variables will be scaled internally. If \fImode\fP = 2,
the scaling is specified by the input \fIdiag\fP. Other
values of mode are equivalent to \fImode\fP = 1.

\fIfactor\fP is a positive input variable used in determining the
initial step bound. This bound is set to the product of \fIfactor\fP
and the euclidean norm of \fIdiag\fP*\fIx\fP if the latter is
nonzero, or else to \fIfactor\fP itself. In most cases factor should
lie in the interval (.1,100.). 100. is a generally recommended
value.

\fInprint\fP is an integer input variable that enables controlled
printing of iterates if it is positive. In this case, fcn is called
with \fIiflag\fP = 0 at the beginning of the first iteration and
every \fInprint\fP iterations thereafter and immediately prior to
return, with \fIx\fP and \fIfvec\fP available for printing. If
\fInprint\fP is not positive, no special calls of fcn with
\fIiflag\fP = 0 are made.

\fIinfo\fP is an integer output variable. If the user has
terminated execution, info is set to the (negative)
value of iflag. See description of fcn. Otherwise,
info is set as follows.

  \fIinfo\fP = 0  improper input parameters.

  \fIinfo\fP = 1  both actual and predicted relative reductions
in the sum of squares are at most \fIftol\fP.

  \fIinfo\fP = 2  relative error between two consecutive iterates
is at most \fIxtol\fP.

  \fIinfo\fP = 3  conditions for \fIinfo\fP = 1 and \fIinfo\fP = 2 both hold.

  \fIinfo\fP = 4  the cosine of the angle between fvec and any
column of the Jacobian is at most gtol in absolute value.

  \fIinfo\fP = 5  number of calls to \fIfcn\fP has reached or
exceeded maxfev.

  \fIinfo\fP = 6  \fIftol\fP is too small. No further reduction in
the sum of squares is possible.

  \fIinfo\fP = 7  \fIxtol\fP is too small. No further improvement in
the approximate solution x is possible.

  \fIinfo\fP = 8 \fIgtol\fP is too small. \fIfvec\fP is orthogonal to
the columns of the Jacobian to machine precision.

\fInfev\fP is an integer output variable set to the number of
calls to \fIfcn\fP.

\fIfjac\fP is an output \fIm\fP by \fIn\fP array. The upper \fIn\fP by
\fIn\fP submatrix of \fIfjac\fP contains an upper triangular matrix
\fIr\fP with diagonal elements of nonincreasing magnitude such that

         t     t           t
        p *(jac *jac)*p = r *r,

where \fIp\fP is a permutation matrix and \fIjac\fP is the final
calculated Jacobian. column \fBj\fP of \fIp\fP is column
\fIipvt\fP(\fBj\fP) (see below) of the identity matrix. The lower
trapezoidal part of \fIfjac\fP contains information generated during
the computation of \fIr\fP.

\fIldfjac\fP is a positive integer input variable not less than
\fIm\fP which specifies the leading dimension of the array
\fIfjac\fP.

\fIipvt\fP is an integer output array of length \fIn\fP. \fIipvt\fP
defines a permutation matrix \fIp\fP such that \fIjac\fP*\fIp\fP =
\fIq\fP*\fIr\fP, where \fIjac\fP is the final calculated Jacobian,
\fIq\fP is orthogonal (not stored), and \fIr\fP is upper triangular
with diagonal elements of nonincreasing magnitude.  Column \fBj\fP
of \fIp\fP is column \fIipvt\fP(\fBj\fP) of the identity matrix.

\fIqtf\fP is an output array of length \fIn\fP which contains
the first \fIn\fP elements of the vector (\fIq\fP transpose)*\fIfvec\fP.

\fIwa1\fP, \fIwa2\fP, and \fIwa3\fP are work arrays of length \fIn\fP.

\fIwa4\fP is a work array of length \fIm\fP.
.br
.SH SEE ALSO
.BR lmder (3),
.BR lmder1 (3),
.BR lmstr (3),
.BR lmstr1 (3).
.br
.SH AUTHORS
Jorge More', Burt Garbow, and Ken Hillstrom at Argonne National Laboratory.
This manual page was written by Jim Van Zandt <jrv@debian.org>,
for the Debian GNU/Linux system (but may be used by others).
